The Power of Abstraction

One of the reasons mathematics is powerful and useful is that it is abstract. A collection of abstract symbols may seem sterile, but the power is in the hands of the practitioner, for you can give the symbols whatever meanings you please.

Consider Boolean algebra, an abstract system of rules for symbolic expressions. As an abstract algebraic system, it just is, end of story. However, if we interpret the “variables” in the system (i.e., the placeholders) as logical variables, which might be true or false, then the rules of Boolean algebra correspond exactly to the rules of classical logic (propositional calculus). The binary operations of the Boolean algebra then correspond to the logical connectives “and” and “or,” and the unary operation corresponds to “negation.”

Alternatively, if we interpret the variables of the system as sets, then the rules of Boolean algebra correspond exactly to the rules of set theory. The binary operations in this case correspond to “intersection” and “union” of sets, and the unary operation corresponds to “complementation.”

Still another type of system that can be represented by Boolean algebra is digital electronic circuits, which are used in electronic computers. In the simplest interpretation in terms of switches (that can either on or off), the binary operations correspond to series and parallel connections, and the unary operation corresponds to changing the state of a switch (from on to off, or off to on). Using circuit elements called logic gates, the binary and unary operations each correspond to different logic gates.

The wonderful thing about having several different perspectives on the same abstract system is that you can use one of the concrete representations to provide insight into the others. Similarly, if you are trying to solve a problem in one of the system, a strategy is to switch to one of the other representations to see if that helps.

For example, if you are designing a digital circuit, and you have one that does the job but you now wish to optimize it, one option is to write a Boolean expression that corresponds to the circuit, and then use the rules of Boolean algebra to simplify the expression. In the process of simplification, you might find it useful to interpret the variables as sets, and then draw Venn diagrams. (Humans typically find visuals helpful.) Once you have simplified the expression, you can then construct a simplified circuit that does the same job that the more complex one did, perhaps saving time and money, and perhaps improving reliability.

Another class of examples comes from differential equations. The second order linear differential equations that describe mechanical vibrations (in damped harmonic systems) also describe electrical vibrations in “LCR” circuits (the kind of circuits that are used to “tune in” to various radio stations in analog radios).

Besides providing problem-solving tools, and using knowledge of one type of system to help provide insight into another, the fact that diverse phenomena can be described by the same mathematics indicates an underlying unity. Beautiful!

This is a primary motivation for learning some mathematics: It provides another avenue for admiring the beauty of nature.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)